施主,你是不是把每一项的分子分母搞颠倒了?如果没有,这题就没法求,除非给定具体的n,然后通分,别无二法.
如果把每项的分子分母颠倒过来,就有办法求和了.
an=[(n+1)^2+1]/[(n+1)^2-1]
=1+2/[n(n+2)]
=1+1/n-1/(n+2)
所以,Sn=(1+1-1/3)+(1+1/2-1/4)+(1+1/3-1/5)+.+[1+1/n-1/(n+2)]
=n+1+1/2-1/(n+1)-1/(n+2)
=n+3/2-(2n+3)/[(n+1)(n+2)]
=n(n+3)(2n+3)/[(n+1)(n+2)]